# Full text of "Treatise On Analysis Vol-Ii"

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```202       XIII    INTEGRATION

spaces «>fc(X; K) (with respect to the norm ||/||); in other words, the re-
striction of // to <>f C(X) is a (complex) measure v. Furthermore, it follows from
the inequality \u(f)\ g c - N^/), where /e Jfc(X), and from (13.3.2.1) that
|v|(|/|) ^ c • N^/) = c - Xl/l). Hence we deduce from (13.15.1) and (13.15.3)
that there exists a locally /Mntegrable function g0 such that \g0(x)\ ^ c almost
everywhere with respect to /j., and such that |v| = g0 - JJL. If v = h - |v| (13.16.3)
we have therefore v = g • fi, where g = g0h (13.14.5), and it is clear that
g e &% '(//). This being so, the linear forms u and 6g agree on the subspace
JTC(X), which is dense in &lc (13.11.6); since they are both continuous on
JS?£, it follows that they are equal (3.15.2). Finally, the fact that Qgi = 602
implies g^ = g2 follows from (13.15.3).                                                Q.E.D.

It follows that the mapping g\-+9gisa linear isometry of the Banach space
L£ onto the dual of the Banach space L£ (12.15).

We remark also that it follows from (13.12.5) that, for each function
/eJS?c(X, ft), the linear form g^-ifgdii on «£?£p(X, //) is continuous with

respect to the topology defined by the seminorm N^ . But in general there
exist linear forms on JS?<? (X, /A) which are continuous with respect to this
topology but are not of the above type.

PROBLEMS

1.    (a)   Let X be a locally compact space, /z a positive measure on X. Let/be a fi-measur-

able mapping of X into R and p a real number such that .1 <^p <| -h oo. Show that

r*
Np(/) = sup       \fg\ d^y where g runs through the set of functions belonging to «^*R(X)

such that Nq(#) g 1, where g = p/(p — 1) (and q = -f oo if p — 1). State and prove an
analogous result for a ft-measurable mapping of X into C.

(b) Suppose that 1 < p< -f oo. For each function # e ^l(X, p) (resp. ^ e JSfi(X, JLI)),
let^J denote the linear form f\~->^(Jg) on L|(X, /*) (resp. L£(X, ft)). Show that
5 »-»• 6g is an isometry of L&(X, /z) (resp. L^(X, jit)) onto the Banach space (5.7.3)
dual to the Banach space L|(X, /z) (resp. Lfc(X, /x)). (Imitate the proof of (13.17.1),
using (a),)

2.   Let X be a compact space, p, a positive measure on X, and (/,) an orthonormal sequence
in xyyi,   ). Write

K.(*. r) - EAWAW^       H.(J) = f |KW(5,

*=i                              Jx

(the "//th Lebesgue function" of the system (/„), cf. Section 11.6, Problem 2). For any
g e ^(X, ft), let JB(^) denote the functionnce of complex measures on X, there
```